System and method for optimizing the operation of an oversampled discrete Fourier transform filter bank

ABSTRACT

A system for, and method of optimizing an operation of an oversampled filter bank and an oversampled discrete Fourier transform (DFT) filter bank designed by the system or the method. In one embodiment, the system includes: (1) a null space generator configured to produce a basis of a null space of a perfect reconstruction condition matrix based on a first window of the oversampled filter bank and (2) an optimizer associated with the basis generator and configured to employ the null space and an optimization criterion to construct a second window of the oversampled filter bank.

CROSS-REFERENCE TO RELATED APPLICATION

This application is based on and claims priority of U.S. ProvisionalPatent Application Ser. No. 60/703,744, filed on Jul. 29, 2005, byMansour, entitled “Optimization Criteria in Oversampled DFT FilterBanks,” commonly assigned with this application and incorporated hereinby reference.

TECHNICAL FIELD OF THE INVENTION

The invention is directed, in general, to digital filters and, moreparticularly, to a system and method for optimizing the operation of anoversampled discrete Fourier transform (DFT) filter bank.

BACKGROUND OF THE INVENTION

Modulated filter banks have been a fundamental tool in many signalprocessing applications (e.g., data compression and subband adaptivefiltering), mainly because they can be efficiently implemented as apolyphase digital finite impulse response (FIR) filter using thewell-known discrete cosine transform (DCT) or discrete Fourier transform(DFT) for signal transformation. Modulated filter banks have beenadopted in modern multimedia standards, including the Moving PictureExperts Group (MPEG) standard (see, e.g., http://www.mpeg.org), which isone of the most successful signal processing schemes in use today.

Critically sampled modulated filter banks have in general proven to bewell-suited for signal compression applications in which processing ofsubband samples involves only quantizing coefficients, since thatminimal amount of processing does not significantly increase aliasingbetween adjacent bands. However, for applications that require extensivesubband sample processing, e.g., subband adaptive filtering (see Chapter7 of Haykin, “Adaptive Filter Theory,” 4th edition, Prentice Hall,2002), and subband dynamic range compression (see, Brenna, et al., “AFlexible Interbank Structure for Extensive Signal Manipulations inDigital Hearing Aids,” IEEE International Symposium on Circuits andSystems (ISCAS), vol. 6, pp. 569–572, 1998), oversampling is necessaryto mitigate aliasing.

Modulated filter banks commonly operate with two windows: an analysiswindow and a synthesis window. The relative placement and shape of thesewindows determine the operation of the modulated filter bank. Portnoff,“Time-Frequency Representation of Digital Signal and Systems Based onShort-Time Fourier Analysis,” IEEE Transactions on Acoustics, Speech,and Signal Processing, Vol. ASSP-28, No. 1, pp. 55–69, February 1980(incorporated herein by reference), addressed the uniform DFT filterbank and described what is known in the art as “perfect reconstructionconditions” for it. (Those skilled in the art understand that “perfect”is a term of art, and does not mean absolute perfection in thecolloquial sense.) Crochiere, “A Weighted Overlap-Add Method ofShort-Time Fourier Transform,” IEEE Transactions on Acoustics, Speech,and Signal Processing, Vol. ASSP-28, No. 1, pp. 99–102, February 1980,set forth a relatively efficient implementation of a DFT filter bankusing a synthesis window overlap-add technique. Subsequent works (e.g.,Shapiro, et al., “Design of Filters for the Discrete Short-Time FourierTransform Synthesis,” IEEE International Conference on Acoustics,Speech, and Signal Processing (ICASSP), 1985; Shapiro, et al., “AnAlgebraic Approach to Discrete Short-Time Fourier Transform Analysis andSynthesis,” ICASSP, pp. 804–807, 1984; and Bolsckei, et al.,“Oversampled FIR and IIR Filter Banks and Weyl-Heisenberg Frames,”ICASSP, Vol. 3, pp. 1391–1394, 1996) address the design of the synthesiswindow. Although the problem of designing the synthesis window isrelatively old, the techniques disclosed in these works were usually the“minimum-norm” solution. For example, Shapiro, et al., “Design ofFilters . . . ,” and Shapiro, et al., “An Algebraic Approach . . . ,”both supra, proposed a minimum-norm solution for different orders of theanalysis window, when the subband samples are processed, afterformulating the problem as a least-square problem. In Bolsckei, et al.,supra, a frame-theoretic technique was described for the design of thesynthesis window using para-unitary prototypes.

Accordingly, what is needed in the art is a better technique foroptimizing the operation of an oversampled DFT filter bank. What is alsoneeded in the art is a DFT filter bank that has been optimized by thetechnique thereby to yield improved operation.

SUMMARY OF THE INVENTION

To address the above-discussed deficiencies of the prior art, theinvention provides, in one aspect, a system for optimizing an operationof an oversampled filter bank. In one embodiment, the system includes:(1) a null space generator configured to produce a basis of a null spaceof a perfect reconstruction condition matrix based on a first window ofthe oversampled filter bank and (2) an optimizer associated with thenull space generator and configured to employ the basis and anoptimization criterion to construct a second window of the oversampledfilter bank.

In another aspect, the present invention provides a method of optimizingan operation of an oversampled filter bank. In one embodiment, themethod includes: (1) producing a basis of a null space of a perfectreconstruction condition matrix based on a first window of theoversampled filter bank and (2) employing the basis and an optimizationcriterion to construct a second window of the oversampled filter bank.

In yet another aspect, the present invention provides an oversampled DFTfilter bank. In one embodiment, the DFT filter bank includes a pluralityof lowpass filters configured to process an input signal to yield anoutput signal based in part on coefficients that describe a secondwindow designed by: (1) producing a basis of a null space of a perfectreconstruction condition matrix based on a first window of theoversampled filter bank and (2) employing the basis and an optimizationcriterion to construct the second window.

The foregoing has outlined preferred and alternative features of theinvention so that those skilled in the pertinent art may betterunderstand the detailed description of the invention that follows.Additional features of the invention will be described hereinafter thatform the subject of the claims of the invention. Those skilled in thepertinent art should appreciate that they can readily use the disclosedconception and specific embodiment as a basis for designing or modifyingother structures for carrying out the same purposes of the invention.Those skilled in the pertinent art should also realize that suchequivalent constructions do not depart from the spirit and scope of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the invention, reference is nowmade to the following descriptions taken in conjunction with theaccompanying drawing, in which:

FIG. 1 illustrates a schematic diagram of an oversampled filter bank inwhich a downsampling factor is less than the number of bands in thefilter bank and constructed according to the principles of theinvention;

FIG. 2 illustrates a graph of a zero-phase low pass filter having acutoff frequency of 2π/M designed using windowing with a Kaiser window;

FIG. 3 illustrates a graph of reconstruction mean-square error fordifferent orders of the synthesis window for the analysis window of FIG.2;

FIG. 4 illustrates a graph of a minimum-order synthesis window;

FIGS. 5A and 5B respectively illustrate time and frequency responses ofa synthesis window and a minimum-norm synthesis window for L_(f)=56;

FIGS. 6A and 6B respectively illustrate time and frequency responses ofa synthesis window and a minimum-norm synthesis window for L_(f)=96;

FIGS. 7A and 7B respectively illustrate least-square synthesis windowscorresponding to sinusoid and white noise templates;

FIG. 8 illustrates a flow diagram of one embodiment of a method ofoptimizing the operation of an oversampled filter bank carried outaccording to the principles of the invention;

FIG. 9 illustrates a graph of reconstruction signal-to-noise ratio (SNR)for rounding and minimum quantization error solutions; and

FIG. 10 illustrates a block diagram of one embodiment of a system foroptimizing the operation of an oversampled filter bank implemented in adigital signal processor (DSP) and constructed according to theprinciples of the invention.

DETAILED DESCRIPTION

As those skilled in the pertinent art understand and as stated above,oversampled filter banks have an analysis window and a synthesis window.The analysis window is usually designed to meet one or more predefinedcriteria pertaining to the use to which the filter bank is to be put.The synthesis window is almost always designed based on the design ofthe analysis window. A novel, general formulation by which the synthesiswindow may be designed given an analysis window will be introducedherein. The general formulation uses the null space of a perfectreconstruction condition matrix to determine the optimal synthesiswindow design and contrasts sharply with conventional techniques, whichare uniformly directed to the “minimum-norm” solution of the synthesiswindow. Different optimization criteria of the synthesis window will bedisclosed, and explicit, closed-form solutions of the differentoptimization criteria will be set forth.

The focus herein will be on a synthesis window design technique thatsatisfies perfect reconstruction conditions. The algebraic approachdescribed in Portnoff, supra, and Shapiro, et al., “Design of Filters .. . ,” supra, will be used herein rather than the frame-theoreticapproach, as it is more illustrative of the optimization objectivefunctions that will be set forth below.

Oversampling yields extra degrees of freedom that can be exploited indifferent ways. In particular, oversampling can produce a non-zero nullspace of the perfect reconstruction condition matrix. This null spacecan be exploited to optimize the objective function of differentcriteria without sacrificing the perfect reconstruction. Closed-formsolutions of the synthesis window design will be described for thefollowing optimization criteria:

1. Minimum delay design. Minimum delay design produces the smallestoutput lag of the causal implementation of synthesis window andrepresents the minimum-order filter that requires the least computationresources.

2. Minimum out-of-band design. Minimum out-of-band design produces thebest frequency selectivity.

3. Least-square design. Least-square design produces the closestsynthesis window to a given template.

4. Minimum quantization-error design. Minimum quantization-error designminimizes the quantization error for a fixed-point implementation.

The perfect reconstruction conditions will first be reviewed. Then, ageneral optimization model for the synthesis window using theminimum-norm solution and the basis of the null space of the perfectreconstruction condition matrix will be described. Next, the differentoptimization criteria will be set forth and an example given for each ofthe criteria using a template low pass analysis window.

Optimization Model

FIG. 1 illustrates a schematic diagram of an oversampled filter bank 100in which a downsampling factor D is less than a number M of bands (e.g.,110, 120, 130) in the filter bank and constructed according to theprinciples of the invention. The filter bank 100 is a so-called “perfectreconstruction, or PR, filter bank, meaning that the filter bank 100operates by decomposing an input signal X(z) 140 by filtering andsubsampling and then reconstructing to yield an output signal{circumflex over (X)}(z) 150 by inserting zeroes, filtering andsummation. For a discussion of PR filter banks in general, see, e.g.,http://cas.ensmp.fr/˜chaplais/Wavetour_presentation/filtres/Perfect_Reconstruction.html.

If H(z) and F(z) represent prototype analysis and synthesis filters(respectively), the k^(th) band analysis and synthesis filters are:H _(k)(z)=H(zW _(M) ^(k)), andF _(k)(z)=F(zW _(M) ^(k)),i.e., h_(k)(n)=h(n)·W_(M) ^(−kn) and f_(k)(n)=f(n)·W_(M) ^(−kn),assuming that support(h(n))ε[−L_(h),L_(h)] andsupport(f(n))ε└−L_(f),L_(f)┘. The reconstructed signal {circumflex over(X)}(z) has the form:

$\begin{matrix}{{{\hat{X}(z)} = {\frac{1}{D}{\sum\limits_{l = 0}^{D - 1}{{A_{l}(z)} \cdot {X( {zW}_{D}^{1} )}}}}},} & (1)\end{matrix}$where

$\begin{matrix}{{{A_{l}(z)} = {\sum\limits_{k = 0}^{M - 1}{{H( {z \cdot W_{D}^{l} \cdot W_{M}^{k}} )} \cdot {F( {z \cdot W_{M}^{k}} )}}}},\mspace{14mu}{{{with}\mspace{14mu} l} = 0},1,\ldots\mspace{14mu},{D - 1.}} & (2)\end{matrix}$

According to Portnoff, supra, the necessary and sufficient conditionsfor perfect reconstruction are:

1. Perfect reconstruction condition:

$\begin{matrix}{{{\sum\limits_{l = {- \infty}}^{\infty}{{h( {{lD} - n} )} \cdot {f( {n - {lD}} )}}} = {1/M}},\mspace{14mu}{{{for}\mspace{14mu} 0} \leq n \leq {D - 1}},\mspace{14mu}{and}} & (3)\end{matrix}$

2. Aliasing canceling condition:

$\begin{matrix}{{{ {{\sum\limits_{l = {- \infty}}^{\infty}{hlD}} - n + {rM}} ) \cdot {f( {n - {lD}} )}} = 0},\mspace{14mu}{{{for}\mspace{14mu} 0} \leq n \leq {D - 1.}}} & (4)\end{matrix}$

In the following, different (yet equivalent) perfect reconstructionconditions will be used that are more illustrative. From Equations (1)and (2), the perfect reconstruction conditions can be derived as:A _(l)(z)=δ[l]This condition can be expressed as:

$\begin{matrix}{{{\sum\limits_{m = {- L_{f}}}^{L_{f}}{{f(m)} \cdot {h( {- m} )}}} = \frac{D}{M}},} & (5)\end{matrix}$

${{\sum\limits_{m = {- L_{f}}}^{L_{f}}{{f(m)} \cdot {h( {- m} )}}} = 0},\mspace{14mu}{{{for}\mspace{14mu} 1} \leq {r} \leq \lfloor \frac{L_{h} + L_{f}}{M} \rfloor},{{\sum\limits_{m = {- L_{f}}}^{L_{f}}{{\cos( \frac{2\;\pi\;{lm}}{D} )} \cdot {f(m)} \cdot {h( {{rM} - m} )}}} = 0},\mspace{14mu}{{{for}\mspace{14mu} 1} \leq l \leq \frac{D}{2}},\mspace{14mu}{{{and}\mspace{14mu} 1} \leq {r} \leq \lfloor \frac{L_{h} + L_{f}}{M} \rfloor},\mspace{14mu}{and}$${{\sum\limits_{m = {- L_{f}}}^{L_{f}}{{\sin( \frac{2\;\pi\;{lm}}{D} )} \cdot {f(m)} \cdot {h( {{rM} - m} )}}} = 0},\mspace{14mu}{{{for}\mspace{14mu} 1} \leq l \leq {\frac{D}{2} - 1}},\mspace{14mu}{{{and}\mspace{14mu} 1} \leq {r} \leq \lfloor \frac{L_{h} + L_{f}}{M} \rfloor},$in other words:

${{\sum\limits_{m = {- L_{f}}}^{L_{f}}{W_{D}^{- {lm}} \cdot {f(m)} \cdot {h( {{rM} - m} )}}} = {\frac{D}{M}{\delta( {l,r} )}}},$for 0≦l≦D−1, and

$1 \leq {r} \leq {\lfloor \frac{L_{h} + L_{f}}{M} \rfloor.}$

It is straightforward to show the equivalence of the two perfectreconstruction conditions. From Equation (5), a total of

$D \cdot ( {1 + {2 \cdot \lfloor \frac{L_{h} + L_{f}}{M} \rfloor}} )$conditions exist. Some of the conditions (at the filter edges) may bedependent. The perfect reconstruction conditions can be put in a matrixform as:H·f=z,  (6)where f=[f(−L_(f)), f(−L_(f)+1), . . . , f(0), . . . , f(L_(f))] is thesynthesis window, H_(D(1+2·└(L) _(h) _(+L) _(f) _()/M┘)×(2L) _(f) ₊₁₎ isthe conditions matrix (which is constructed from the conditions inEquation (5)), z=[D/M,0,0, . . . ,0]. Note that:

$\begin{matrix}{{{rank}(H)} \leq {D \cdot {( {1 + {2\lfloor \frac{L_{h} + L_{f}}{M} \rfloor}} ).}}} & (7)\end{matrix}$

The minimum-norm solution f^(#) is calculated as:f ^(#) =H ^(#) ·z,  (8)where H^(#) is the pseudo-inverse of the condition matrix H (see, e.g.,Chapter 5 of Golub, et al., Matrix Computation, 3rd edition, The JohnHopkins University Press, 1996). The dimension of the null space of His:K=2L _(f)+1−rank(H).

If the basis of the null space is {v _(i)}_(i=1:K), a general formulafor a synthesis window f that satisfies the perfect reconstructionconditions is:

$\begin{matrix}{{\underset{\_}{f} = {{\underset{\_}{f}}^{\#} + {\sum\limits_{i = 1}^{K}\;{C_{i} \cdot {\underset{\_}{\nu}}_{i}}}}},} & (9)\end{matrix}$where {c _(i)}_(i=1:K) are scalars. Equation (9) represents the generalmodel for optimization that will be used below.

No vectors in the null space of H interfere with the perfectreconstruction conditions set forth in Equation (5). Therefore, anyvector v in the null space satisfies the following condition:

$\begin{matrix}{{{\sum\limits_{m = {- L_{f}}}^{L_{f}}\;{W_{D}^{- {lm}} \cdot {\nu(m)} \cdot {h( {{rM} - m} )}}} = 0},\mspace{14mu}{{{for}\mspace{14mu} 0} \leq l \leq {D - 1}},\mspace{14mu}{{{and}\mspace{14mu} 0} \leq {r} \leq {\lfloor \frac{L_{h} + L_{f}}{M} \rfloor.}}} & (10)\end{matrix}$

No structure for h is assumed in the above discussion. In fact, even ifh is an arbitrary vector, the above perfect reconstruction formula stillapplies. However, in most practical applications, h is a lowpass filterhaving a bandwidth of 2π/M

Note that in the perfect reconstruction conditions set forth in Equation(5), h and f may be exchanged without affecting the perfectreconstruction, i.e., the synthesis window may be designed first andthen the analysis window chosen that satisfies the perfectreconstruction condition using substantially the same procedure.

Optimization Criteria

Various optimization criteria for constructing a synthesis window f thatsatisfies Equation (9) will now be described. With each criterion, theobjective is to find {c _(i)}_(i=1:K) in Equation (9) that optimize acertain objective function. This optimization is possible whenever anonzero null space of the condition matrix H exists. Redundancy may beexploited by reducing the order of the synthesis window as describedbelow rather than optimizing its design. The optimization examplesdescribed herein may be extended to other filter design optimizationsusing the model described in Equation (9).

To illustrate the different optimization criteria better, a designexample will be included with each criterion. In all the examples, thefollowing filter bank parameter settings will be used: M=32, D=8 andL_(h)=64. The analysis window is as shown in FIG. 2, namely a zero-phaselow pass filter curve 200 having a cutoff frequency of 2π/M that isdesigned using windowing with a Kaiser window.

Least-Order Solution

The smallest order of the synthesis window is directly related to therank of the condition matrix H. If the synthesis window order equalsrank(H), H is a full rank square matrix and we have only one solution,i.e., the redundancy is exploited by reducing the order of the windowrather than optimizing the design as in Equation (9). Therefore theleast order of the synthesis window satisfies the condition:

$\begin{matrix}{{{2{\hat{L}}_{f}} + 1} = {{{rank}(H)} \leq {D \cdot {( {1 + {2\lfloor \frac{L_{h} + {\hat{L}}_{f}}{M} \rfloor}} ).}}}} & (11)\end{matrix}$

{circumflex over (L)}_(f) is evaluated and approximated to the nearestupper integer (which may result in increasing the dimension of the nullspace of H to one). After approximation, rank(k)=2{circumflex over(L)}_(f)+1, H^(#)=H⁻¹, and the null space of H contains only the zerovector. Longer synthesis window results in larger null space dimensions,which allows more degrees of freedom. All subsequent optimizationcriteria assume that the synthesis window order is larger than{circumflex over (L)}_(f).

If the order is less than {circumflex over (L)}_(f), perfectreconstruction is no longer possible. However, the least-square solutionof Equation (6) yields the best possible reconstruction for a givenorder. FIG. 3 shows a reconstruction mean-square error curve 300 fordifferent orders of the synthesis window for the analysis window of FIG.2 where {circumflex over (L)}_(f)=20.

For the minimum-order synthesis window,

${{{rank}(H)} \leq {D \cdot ( {1 + {2\lfloor \frac{L_{h} + L_{f}}{M} \rfloor}} )}} = 40.$FIG. 4 shows a synthesis window 400. In this case, the dimension of thenull space is one (because of the upper integer approximation).

For causal systems, the least order solution is equivalent to theminimum delay solution, a desirable feature in many real-timeapplications, e.g., subband adaptive filtering.

Minimum-Norm Solution

Note that {v _(i)}_(i=1:M) in Equation (9) are orthogonal to the columnsof H^(#). As a result:

${\underset{\_}{f}} = {{{\underset{\_}{f}}^{\#}}^{2} + {\sum\limits_{i = 1}^{K}\;{C_{i}^{2}.}}}$Therefore, the minimum-norm solution is {circumflex over (f)}=f ^(#).

C. Minimum Out-of-Band Energy Solution

If a lowpass filter prototype is used, the bandwidth of theanalysis/synthesis window is 2π/M . If the DFT of f is denoted byf(e^(jω)), the objective function becomes:

$\begin{matrix}{\min{\int_{2{\pi/M}}^{\pi}{{{F( {\mathbb{e}}^{j\;\omega} )}}^{2} \cdot \ {{\mathbb{d}\omega}.}}}} & (12)\end{matrix}$

If the DFT of each of the base vectors v _(i) is V _(i), then Equation(9) yields the following:

$\underset{\_}{F} = {{\underset{\_}{F}}^{\#} + {\sum\limits_{i = 1}^{K}\;{c_{i} \cdot {{\underset{\_}{V}}_{i}.}}}}$

Assuming the length of the DFT is N, only the first N/2+1 components areretained (because of the real symmetry), i.e., F, F ^(#), and {V _(i)}are arrays of size N/2+1.

If A is a diagonal matrix of size (N/2+1)×(N/2+1) such that a_(i,i)=0for

$i < \frac{N}{M}$and a_(i,i)=1 other wise V_((N/2+1)×K) is a matrix that has {V _(i)} asits columns and c=[c₁, . . . , c_(M)]^(T), the objective function can beapproximated as:J( c )= F ^(H) ·A·F =( F ^(#) +V _(c) )^(H) A·( F ^(#) +V _(c) ), and= F ^(#H) ·A·F ^(#) +b ^(H) ·c+c ^(H) ·b+c ^(H) ·G.c.where b=V^(H) AF ^(#) and G=V^(H) AV. Differentiating with respect to cyields:

$\frac{\partial J}{\partial c^{*}} = {b + {G \cdot {\underset{\_}{c}.}}}$Therefore the optimal solution in this case is:ĉ=−(V ^(H) AV)⁻¹ ·V ^(H) AF ^(#).  (13)

Note that V^(H) AV is K×K matrix, and a necessary condition for V^(H) AVto have full rank is

${\frac{N}{2} + 1 - \frac{N}{M}} > {K.}$

The minimum out-of-band power design will be illustrated by two exampleswith L_(f)=56 and L_(f)=96, respectively. FIGS. 5A and 5B show the timeand frequency responses of the synthesis windows compared with theminimum-norm synthesis window for L_(f)=56, and FIGS. 6A and 6B show thetime and frequency responses of the synthesis windows compared with theminimum-norm synthesis window for L_(f)=96. FIGS. 5A, 5B, 6A and 6B showa reduction of more than 30 dB in the out-of-band power. Thissignificant reduction results in a better frequency-selective window, adesirable feature in practical applications in which the original signalis usually contaminated with noise.

Least-Square Solution

In this case, the objective is to minimize the difference between thesynthesis window and a window template g, i.e., the objective functionis:min∥f−g∥².  (14)If:Λ_((2L+1)×K)=( v ₁ ,v ₂ , . . . ,v _(K)),  (15)the cost function has the form:J( c )=( f ^(#) +Λc−g )^(H)·( f ^(#) +Λc−g ).Therefore:

$\frac{\partial J}{\partial c^{*}} = {{\Lambda^{H}( {{\underset{\_}{f}}^{\#} - \underset{\_}{g}} )} + {\Lambda^{H}\Lambda{\underset{\_}{c}.}}}$Hence, the optimal solution in this case is:ĉ=−(Λ^(H)Λ)⁻¹.Λ^(H)( f ^(#) −g ).

By noting that Λ^(H) Λ=I and Λ^(H)·f ^(#)=0 (because f ^(#) is in therow space of H), the above relation can be simplified to:ĉ=−Λ ^(H) ·g.  (16)

To illustrate the significance of the least-square solution, thetemplate g is chosen to be significantly different from f ^(#) in twoextreme examples. In the first example, g is chosen to be a sine wave.In the second example, g is chosen to be an arbitrary sequence. Theresulting synthesis windows are illustrated in FIGS. 7A and 7B. In bothexamples, L_(f)=96.

Minimum Quantization-Error Solution

In practical applications, the synthesis window will be implementedusing finite-precision arithmetic. Therefore, coefficients quantizationerror is an important design criterion. The best quantization of eachcomponent in f (to the upper or lower integer) such that the overallquantization error is minimized will now be considered. In other words,the best quantized solution in the neighborhood of an existing solutionf will be examined. The objective function in this case is:min∥Q(f)∥²,  (17)where Q(·) is the quantization error function defined as:Q( f )= f+Λ·c −(└ f┘+α ),  (18)and where └·┘ is the floor integer function, Λ is as defined in Equation(15), α is an (2L_(f)−1) binary vector that represents the quantizationapproximation, i.e., if Q(f_(i))=└f_(i)┘, then α_(i)=0, and ifQ(f_(i))=┌f_(i)┐, then α_(i)=1. Note that Equation (18) contains twounknown vectors, c and α. If e denotes the initial quantization error,then:e=f−|f|.The objective function can be written in the form:min∥Λ·c+(e−α)∥².  (19)

For any value of α, the value of c that yields the least square solutionof Equation (19) is:ĉ=−Λ ^(H)·( e·α ).For this reason, the objective function of Equation (19) can be writtenin the form:(e−α)^(H)·P·(e−α),  (20)where P=(I−ΛΛ^(H))^(H)(I−ΛΛ^(H)). This is an integer programming problemwith a quadratic objective function. A simulated annealing technique(see, e.g., Chapter 12 of Rardin, Optimization in Operations Research,Prentice Hall, 1998, incorporated herein by reference) may be used tosolve it. The simulated annealing technique Rardin discloses has thefollowing seven steps:

1. Starting with a feasible solution α ⁽⁰⁾, compute the correspondingvalue of the objective function. Set this value as the incumbentsolution {circumflex over (α)} and the pivot solution {tilde over (α)}.

2. Define a set Ω, of all feasible moves that lead to another feasiblesolution.

3. Arbitrarily choose a move Δα ^((t))εΩ, and compute the new solution α^((t))={tilde over (α)}+Δα ^((t)), and compute the new objectivefunction obj^((t)).

4. If Δα improves the incumbent solution with probability e^((obj)^((t)) ^(−obj) ^((t−1)) ^()/q) (where q is an algorithm parameter), set{tilde over (α)}=α ^((t)).

5. If obj^((t)) is better than the incumbent solution, set {circumflexover (α)}=α ^((t)).

6. Reduce q after a sufficient number of iterations.

7. Repeat for t≦t_(max).

The maximum weight of any move in Ω may be restricted, e.g., to five.The initial solution is set to the rounding solution, i.e., α⁽⁰⁾=|f+0.5|−|f|.

In one embodiment, the simulated annealing technique may be slightlymodified such that once a new optimum is found in step (5), allneighboring solutions within a Hamming distance of one are investigatedbefore arbitrarily selecting from the moving list.

Although the above description is largely directed to a DFT filter bank,the principles of the invention readily apply to other types ofoversampled filter banks. The condition matrix has a different structurebut the redundancy resulting from oversampling can exploited withoutdeparting from the scope of the invention.

FIG. 8 illustrates a flow diagram of one embodiment of a method ofoptimizing the operation of an oversampled filter bank carried outaccording to the principles of the invention. The method begins in astart step. In a step 810, a null space of a perfect reconstructioncondition matrix is produced based on a first window of the oversampledfilter bank. In a step 820, the null space and an optimization criterionare employed to construct a second window of the oversampled filterbank. In a step 830, filter coefficients that describe the first andsecond window are stored in a memory associated with a DSP. The filtercoefficients are then available to be employed in a filter bankassociated with the DSP. The method ends in an end step.

FIG. 9 illustrates the performance of a disclosed embodiment of theinvention versus a rounding solution (which is conventionally used forfixed-point systems). The test data is arbitrary signals with unityvariance. The number of iterations (t_(max)) equals 10⁵, and L_(f)=96.FIG. 9 indicates up to an 18 dB improvement without the need to increaseresolution. This is equivalent to adding three bits of resolution to theDFT filter bank.

FIG. 10 illustrates a block diagram of one embodiment of a system,generally designated 1000, for optimizing the operation of anoversampled filter bank implemented in a DSP, generally designated 1010,and constructed according to the principles of the invention. Theoversampled filter bank includes a plurality of lowpass filters (notshown in FIG. 10, but shown in FIG. 1) configured to process an inputsignal to yield an output signal based in part on coefficients thatdescribe a first window and in part on coefficients that describe asecond window. The first window may be an analysis window, and thesecond window may be a synthesis window.

A null space generator 1020 within the system 1000 is configured toproduce the basis of the null space of a perfect reconstructioncondition matrix based on the first window. An optimizer 1030 isassociated with the null space generator 1020. The optimizer 1030 isconfigured to employ the basis and an optimization criterion toconstruct the second window. The result of this process is a pluralityof filter coefficients that are caused to be stored in a memory 1040associated with the DSP 1010. The filter coefficients are then madeavailable from the memory 1040 to an oversampled filter bank 1050 duringoperation of the DSP 1010.

Although the invention has been described in detail, those skilled inthe pertinent art should understand that they can make various changes,substitutions and alterations herein without departing from the spiritand scope of the invention in its broadest form.

1. A system for optimizing an operation of an oversampled filter bank,comprising: a null space generator configured to produce a basis of anull space of a perfect reconstruction condition matrix based on a firstwindow of said oversampled filter bank; and an optimizer associated withsaid null space generator and configured to employ said basis and anoptimization criterion to construct a second window of said oversampledfilter bank.
 2. The system as recited in claim 1 wherein saidoversampled filter bank is an oversampled discrete Fourier transformfilter bank.
 3. The system as recited in claim 1 wherein saidoversampled filter bank comprises lowpass filters.
 4. The system asrecited in claim 1 wherein said first window is an analysis window andsaid second window is a synthesis window.
 5. The system as recited inclaim 1 wherein said optimization criterion is selected from the groupconsisting of: a least-order design criterion, a minimum-norm designcriterion, a minimum out-of-band energy design criterion, a least-squaredesign criterion, and a minimum quantization-error design criterion. 6.The system as recited in claim 1 wherein said optimizer employs asimulated annealing technique to construct said second window.
 7. Thesystem as recited in claim 1 wherein said second window f has a generalformula$\underset{\_}{f} = {{\underset{\_}{f}}^{\#} + {\sum\limits_{i = 1}^{K}\;{c_{i} \cdot {\underset{\_}{v}}_{i}}}}$and said optimizer applies said optimization criterion to {c_(i)}_(i=1:K).
 8. A method of optimizing an operation of an oversampledfilter bank, comprising: producing a basis of a null space of a perfectreconstruction condition matrix based on a first window of saidoversampled filter bank; and employing said basis and an optimizationcriterion to construct a second window of said oversampled filter bank.9. The method as recited in claim 8 wherein said oversampled filter bankis an oversampled discrete Fourier transform filter bank.
 10. The methodas recited in claim 8 wherein said oversampled filter bank compriseslowpass filters.
 11. The method as recited in claim 8 wherein said firstwindow is an analysis window and said second window is a synthesiswindow.
 12. The method as recited in claim 8 wherein said optimizationcriterion is selected from the group consisting of: a least-order designcriterion, a minimum-norm design criterion, a minimum out-of-band energydesign criterion, a least-square design criterion, and a minimumquantization-error design criterion.
 13. The method as recited in claim8 wherein said employing comprises employing a simulated annealingtechnique to construct said second window.
 14. The method as recited inclaim 8 wherein said second window f has a general formula$\underset{\_}{f} = {{\underset{\_}{f}}^{\#} + {\sum\limits_{i = 1}^{K}\;{c_{i} \cdot {\underset{\_}{v}}_{i}}}}$and said employing comprises applying said optimization criterion to {c_(i)}_(i=1:K).
 15. An oversampled discrete Fourier transform (DFT)filter bank, comprising: a plurality of lowpass filters configured toprocess an input signal to yield an output signal based in part oncoefficients that describe a second window designed by: producing abasis of a null space of a perfect reconstruction condition matrix basedon a first window of said oversampled filter bank, and employing saidbasis and an optimization criterion to construct said second window. 16.The DFT filter bank as recited in claim 15 wherein said first window isan analysis window and said second window is a synthesis window.
 17. TheDFT filter bank as recited in claim 15 wherein said optimizationcriterion is selected from the group consisting of: a least-order designcriterion, a minimum-norm design criterion, a minimum out-of-band energydesign criterion, a least-square design criterion, and a minimumquantization-error design criterion.
 18. The DFT filter bank as recitedin claim 15 wherein said optimizer employs a simulated annealingtechnique to construct said second window.
 19. The DFT filter bank asrecited in claim 15 wherein said second window f has a general formula$\underset{\_}{f} = {{\underset{\_}{f}}^{\#} + {\sum\limits_{i = 1}^{K}\;{c_{i} \cdot {\underset{\_}{v}}_{i}}}}$and said optimizer applies said optimization criterion to {c_(i)}_(i=1:K).
 20. The DFT filter bank as recited in claim 15 whereinsaid DFT filter bank is implemented in a digital signal processor.